A real variable definition of the hilbert transform on d' and s'

Abstract

We define the Hilbert transform Tf of a distribution f as a series in which each term is the product of the Hilbert transiormol an adeouate element of E′ and a power of real variable. A property of some connnutators involving this transform is proved which implies that the image of any real entire function is a real entire function. We next introduce a specific definition for the Hilbert transform on J′. The latter differs from Tf by a real entire function and satisfies some easily applicable properties. it coincides on BMO with the Hilbert Transform considered by C. Fefferman. The definitions presented here are, in some sense, equivalent to those based on complex analytic representations of distributions.